I am new to this list (although I remember many people from the original NM1 list).

I have a question about Formant FM. In the G2 manual it is mentioned that if you modulate a signal of 0 Hz you get all sorts of mysterious waveforms. Well, actually if you set an Osc to partial-mode and then turn the knob fully counterclockwise, you will notice that the frequency is not 0 Hz. If you switch to frequency-mode it's close to 0.2 Hz. This is expected as there cannot be a sound of 0 Hz. When you FM this supposedly 0 Hz signal the sidebands will be c+m and c-m with c being the frequency of the carrier and m being the frequency of the modulator. However, c-m frequencies will "fold back" and produce anti-phase frequencies which will cancel the c+m ones.

I've never used this technique before, so I hope some people who have will respond. The formulas you gave, c-m and c+m, are not for FM but for balanced modulation, or multiplication (often called ring modulation). The formulas for FM are a considerably more complicated._________________--Howard
my music and other stuff

The formula you've used is correct for calculating the frequency of a given sideband, but FM contains infinite sidebands of varying amplitudes. My guess is that the amplitudes on one side are vastly different than the ones in the antiphase zone which creates the strange timbres... but I haven't experimented much with this technique before so I could be wrong...

I'm pulling info from my brain that's unused and 15years old, but I'm pretty sure it has to do with the use of a BiPolar signal as the modulation source. If you used a unipolar signal, the sidebands would cancel, but since the signal is Bipolar, the phases add in the useful sideband. I think a similar thing relates AM and Ring Modulation. Have to start digging to verify.

Doesn't electro80 have a song "Blinded by Memory"? I think I can relate. Was it this, or was it that? once doubt creeps in, memory can blind you...damn hypothalmus.

I am new to this list (although I remember many people from the original NM1 list).

I have a question about Formant FM. In the G2 manual it is mentioned that if you modulate a signal of 0 Hz you get all sorts of mysterious waveforms. Well, actually if you set an Osc to partial-mode and then turn the knob fully counterclockwise, you will notice that the frequency is not 0 Hz. If you switch to frequency-mode it's close to 0.2 Hz. This is expected as there cannot be a sound of 0 Hz. When you FM this supposedly 0 Hz signal the sidebands will be c+m and c-m with c being the frequency of the carrier and m being the frequency of the modulator. However, c-m frequencies will "fold back" and produce anti-phase frequencies which will cancel the c+m ones.

So, where does sound comes from?

Yannis

When an oscillator is set to 0 Hz it is like the oscillation has stopped, the output of the oscillator will stay at a fixed level and not go up and down anymore. Then, when applying LinFM or TrkFM the output level will be 'pushed up and down' in the rhythm of the modulation wave. This produces a new wavefrom which will of course inherit the pitch of the modulating waveform. The produced waveform will have the property that it in general exhibits strong formant areas in the timbre. The position of these formant areas in the spectrum is depending on the modulation depth. LinFM will keep the formants more or less in the same positions while TrkFM will track these formant areas along with the pitch.

Basically, the osc set to 0 Hz acts as a lookup table where the set waveform is sort of what is described in this lookup table. The modulating wavefrom acts as the index in this lookup table. So, linear FM has turned into waveshaping here, as the pitch from the modulating wave is the output pitch and will not change, while the newly shaped waveform can be morphed through an enormous amount of shapes, each with relatively easy to control formant areas in its timbre.
To summarise; 0 Hz linear FM is in fact waveshaping where the waveform of the modulating wave is shaped through the combination of the basic waveform on the carrier osc and the modulation depth plus initial phase relations (depends on the phase of the carrier waveform when the carrier was 'stopped' when it was set to 0 Hz, this defines the actual fixed level on the carrier output without modulation and defines the 'lookup table' wrap around behaviour).

This waveshaping view is a much healthier way of looking at it than when using the academic c+m, c-m and Bessel functions stuff. The latter is of course necessary for scientific audio research, but does give little hands-on sense of what sort of sound is actually produced. And with the c-m and c+m you will also have to take into account their phase relations, which depend upon the phase relations between the carrier and the modulator. It is definitely not that c-m and c+m are always in antiphase. These signals being in antiphase is a misconception, if this were true FM would be a lot easier to predict.

If you have difficulty grasping this thing then simply forget the theory and instead explore it by experimenting and listening to the produced sounds. With an oscilloscope application you can optionally check how the waveform changed come about.

When I was studying electronics, we where exposed to FM also.
The two variables are the modulation index (the depth) and the frequency ratio of modulator and carrier.
The spectra of FM sounds can be calculated (with Bessel functions) and what I remember most from the FM spectra, is that they are rich in formants. You could say that Frequency Modulation is creating formant rich specra most of the time....

thanx for all the explanations. Kofi pointed to a very nice article in SOS and the formula in question is:

W(sb) = W(c) +- nW(m)

where sb is the sideband c is the carrier, m is the modulator and n is the number of sidebands. This is precisely the equation I have in my CSound bok. Well, with O Hz you obviously have

w(sb) = +-nW(m)

which is to say that the frequencies of the sidebands will be m and -m, 2m and -2m and so on. To my knowledge there is no such thing as negative frequency. It is just a frequency with its phase inverted. When two signals of the same frequency are in 180 out of phase (as the above are) they will simply cancel each other out.

Again, I don't see why amplitudes of positive sidebands should be greater or less than negative ones. This depends on the modulation index (amplitude/frequency of the modulator). If that changes for positives sidebands it will also change for negative ones. Have I got this wrong?

Rob mentioned that the anti-phase phenomenon is actually a misconception. I suppose that if this is the case, then the whole thing depends on the phase of the modulator at the time FM is applied.

Even the name (Formant FM) leads to an explanation, though an unsatisfactory one. Formants are just regions of high energy concetration in the spectrum. High energy concetration means that some partials are stronger (high Q) while others are non-existent.

A lookuptable is a thing that you enter with an index value and that will return a value it found at that index.

An example is an oscillator set to 0 Hz for which you have a phase control input, the pahse is the index, the 0 Hz oscillator is the table. By varying the phase beteen zero and maximum you can step through the values of the wave form.

Other examples are the shape LFOs and the control sequencer.

When you apply a saw signal to the phase input thes modules will calculate a function for you that happens to be the wave shape of the oscillator,, when the saw goes downwards the wave is played backwards.

In this respect 0 Hz FM is more like wave shaping, because you are playing the oscillator signal forwards, backwards and with different speeds.

This does not mean that the sideband formulas you gave are wrong, but those apply to the spectrum and not to the momentary resulting signal. Damn, I'm talking FM, time to quit :-)

Ehm one more thing, it might be wrong, but acually the mean amplitude spectrom could very well be zero, but the power spectrum probably not.

Rob mentioned that the anti-phase phenomenon is actually a misconception. I suppose that if this is the case, then the whole thing depends on the phase of the modulator at the time FM is applied.

Yannis.

Correct.

You should not see it as 'anti-phase' but perhaps better as 'mirrored phase'. As the c+m and c-m results have a phase that is 'mirrored' around a certain point. Meaning that the actual phase difference between the two results can just be about anything, with the mirror point exactly in the middle.

Let say that when using two sines the one result would be c+m and an average phase shift of +10 degrees in respect to the carrier signal, then the c-m would have a phase shift of probably -10 degrees (though I don't have the formulas at hand to do the exact calculation). Meaning that when c = 0 Hz, you would get two signals at frequency m that are actually 20 degrees out of phase and not 180 degrees. This is the reason why when the c and m are slightly detuned there is a strong phasing-like shifting in the timbre of the FM sound, and not just a static timbre that has an increasing and decreasing amplitude only. Which would happen if the c+m and c-m would always be in exact antiphase.
Note that an unmodulated signal of 0 Hz will have a certain fixed level output value, and the output value is defined by the phase position. The output value of a 0 Hz sine wave would be 0 only if the phase is 0 degrees or 180 degrees. But if it were 90 degrees the output would stay fixed at a level of 1, and at 45 and 135 degrees at roughly 0.7.
All FM modulation would now appear to be distortion of the amplitude level, using the fixed level (defined by the carrier phase) as the amplitude reference level around which to 'swing' up and down. And using a non-linear distortion function based on the waveform of the carrier wave. E.g. if the carrier wave is a sine, the non-linear distortion function is simply a sine function.

To shed some more light on the 'mechanics' behind this thing:
On a waveform there are only two basic directions into which a deformation can be accomplished, the vertical direction, which means that the amplitude is modulated, and the horizontal direction, which means that the 'phase' is modulated. This last thing is simply shifting the waveform forwards and backwards in time. This is as the 'graph' of a waveform is actually a two-dimensional picture. All deformations can be described by combining or analyzing the deformation into the horizontal direction and the deformation into the vertical direction.

Interestingly there are manipulations into the horizontal direction that appear to produce deformations into the vertical direction and vice versa. An example is when frequency modulation produces another timbre, but not another pitch. Zero Hz FM is just an exemple of such a case, as many waveforms can be produced, but always at the pitch of the modulating signal. Meaning that zero Hz FM can also be expressed as 'result = SIN(a times current modulating wave level value plus phase position of carrier)', if a sinewave is used for the carrier. And note that this formula only works on the amplitude level.

If the linear frequency parameter is modulated there will be deformations into both the vertical and the horizontal direction. And if there is a deformation into the horizontal direction it means that there is a phase shift of a certain amount.
So, the thing is that there is symmetry in this thing, this symmetry can e.g. be that all deformations on the waveform are 'line symmetric' around a diagonal line or 'point symmetric' in respect to a certain point in the waveform graph. So, this c-m and c+m mean that there are two results that are symmetric around c, both in frequency AND in phase.

It is just that all explanations of FM that show only the frequency relations formulas, but do not include the phase relations between the two original signals, are limited. They would all probably assume a phase relation of zero degrees, but say nothing about what happens when there is another phase relation. And if the frequency of a waveform is calculated with 24 bit precision this means that in this system the chance to have an exact zero degree phase relation is only 1 in roughly 16 million(!). So, SOS is not necessarily wrong, but utterly incomplete by a factor of 1 out of roughly 16 million when it is about the Nords. As are many other popular descriptions of FM that pretend to show the mathematics in a 'simplified' way. The math is in reality utterly complex, not much fun to delve into. And I think the complete math can also only be understood if it is understood how amplitude and phase are both always part of the deal and never ride alone.

It is similar to a Fourier transform, where one gets a set of data for both sine and cosine components, which must then be combined together to get the actual amplitude and phase of a certain partial.

Imho it should be forbidden to publish about FM with the use of formulas, unless the complete math is included. As it leads to misconceptions, and only very few seem to grasp that it is actually misleading. It also makes it appear like FM is a mathematical thing. In fact, FM it is much more a mechanical thing, a mechanism with a certain behaviour. And it just happens that the behaviour can be described by mathematical formulas, but also in other and more practically useful ways.
It should be clear that by using only half of the formulas necessary to describe the phenomenon, it is impossible to even mathematically explain the full behaviour of the mechanics. And then FM starts to be a 'difficult' technique. Instead, by studying and experimenting with the behaviour it is imho very well possible for everyone to get an idea what FM does in practice. And that something like zero Hz FM does lead to a certain class of sounds, with certain sonic characteristics that are actually audible, and not in utter silence.

It would be much better if these articles would concentrate on a common sense approach towards FM, skipping all math but trying to explain the mechanics instead. As this incomplete 'math flashing' reeks like narcism on part of the writer. A simple reference to Chowning for those interested in the math would imho do much better. And then fully concentrate the article on the musical side of things, we are musicians after all. Not?

The strange thing is that all the SOS articles, except the FM articles, are in general excellent. The FM articles are for me a prime example of how not to explain FM. Which doesn't mean that FM must be difficult when even SOS didn't manage to do it on an acceptable level. It just means that accidentally in the past some sort of smoke screen was put over FM, and everyone is still busy wiping the smoke induced tears out of their eyes.

For myself, in the eighties I tried to apply the complete FM math, trying if given a certain evolving spectrum and a certain DX algorithm it would be possible to calculate the necessary parameter and envelope settings. I didn't succeed and to my knowledge no one really did. Anyway, after wasting much valuable time and energy on this subject I could only conclude that the mathematical way was not the sensible way to go about with FM. As in practice the math just seems pretty useless, let alone incomplete math. My advice is: if you're not an academic with an academic interest in FM, don't bother about the math, it ain't worth it. Instead, tweak tweak and tweak, until you get the 'feel' of FM sounds and how you can use them in your music.

Well, actually if you set an Osc to partial-mode and then turn the knob fully counterclockwise, you will notice that the frequency is not 0 Hz. If you switch to frequency-mode it's close to 0.2 Hz. This is expected as there cannot be a sound of 0 Hz.

Yannis

Note that the scales for Partial mode and Frequency mode do not relate or translate to each other at all. In Partial mode 0 Hz is actually indeed 0 Hz.

It is true that a signal of 0 Hz is inaudible for humans, like most signals below 20 Hz. So, if you say "there cannot be a sound at 0 Hz" it is as true as saying that there cannot be a sound at 2 Hz. But there can be frequencies with these values.

many thanx for your explanation. It is much more clear to me now. Thing is, I always approach techniques both musically and scientifically just to get a better understanding of the inner workings but there are always incomplete and unclarified explanations. I don't mean to point out any book/author in specific, but I guess that it's just their wish to bring some of the maths to people. Obviously, as you mentioned, in doing so they unintentionally made things obscure.

Blue Hell thanx for your explanation as well. Now that you mentioned it, couldn't it be possible to construct a wavesequencing pattern by feeding values sequentially to an output? A couple of constant modules, a sequencer clocked at audio rates and a multiplexer comes to my mind. This trick I got from Serge modulars where you can speed up the sequencer until its pattern becomes "blurred".

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